Geometric and Topological Inference

Let’s start with the title. I know the meaning of the adjectives, and I know what the noun means, but what does the sentence mean?  Inference is a logical notion, so what do geometry and topology have to do with it? Well, what can one infer from geometrical or topological data? Here’s what the authors say: “In many practical situations, geometric objects are only known through a finite set of possibly noisy sample points. A natural question is then to recover the geometry and the topology of the unknown object from this information.”  A little bit later they say that “[t]his book intends to cover various aspects of geometric and topological inference, from data representation and combinatorial questions to persistent homology, an adaptation of homology to point cloud data.” They aim “to describe the mathematical and algorithmic foundations of the subject.”

So, now that we are getting our bearings a little, let’s get down to brass tacks. Here’s what the authors propose to do: “Part III is devoted to the problem of reconstructing a submanifold M of Rd from a finite point sample [set] … The ultimate goal is to compute a triangulation of M, i.e. a simplicial complex that is homeomorphic to M.”  And so we see the propriety of focusing on homology. Indeed, the idea is to use Delauney complexes and sundry other things Delauney (developed in Part II) to get at this problem, and this requires a good deal of topology and geometry in order to get off the ground. The authors are very thorough in preparing the stage for their dénouement. For the proverbial (mature) non-specialist, the book’s Part I is devoted to “Topological Preliminaries,” and here the focus falls, appropriately, on simplicial complexes. That said, it is probably a good idea for the reader not to be a complete neophyte: he or she should have some familiarity with, say, Spanier or May or Hatcher --- you get the picture.

Finally, Part IV deals with “Distance-based Inference” and contains some tantalizing stuff; for example, under the title “homology inference” we get a lot of material on the inner life and computation of Betti numbers. It is here that we encounter so-called persistent homology. Say the authors: “Chapter 11 introduces persistent homology and provides tools to robustly infer the homology of sampled spaces.”

Well, it’s a brave new world, at least for some of us. The questions addressed in this book (and others like it) belong to a world that I, for one, am really quite alien to, featuring the accelerating application of recently ever-so-pure mathematics to problems that have obvious real-world roots. Somewhere G. H. Hardy is frowning. But this business of geometric inference is very hot stuff, and is getting hotter and hotter as computing power increases. This excellent book is peppered with flow-charts that would make Fortran devotees’ mouths water, aiming at such things as “Computing Delauney filtrations” (Algorithm 6, p. 102) and “Betti numbers computation” (Algorithm 9, p. 201); the latter starts with an input of “[a] filtration of a d-dimensional simplicial complex K containing m simplices,” and ends with the output of nothing less than the d Betti numbers of K. Of course this is very beautiful stuff in and of itself, and not least for pedagogical reasons: this flow-chart, or program, reveals the structure and meaning of the simplicial complex as measured in terms of the corresponding characterization of the ranks of the homology groups, dissected.

So it is fair to say that this book scores high marks on a number of counts. Not only does it address very sexy and fecund contemporary material that bridges pure and applied mathematics is a way heretofore hardly imaginable (at least to curmudgeonly fogeys like me), it is of considerable pedagogical use. The reader gets airborne quickly and gets to fly pretty high.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.